Wednesday, September 22, 2010

Let us examine the mathematics behind this. We will find that they are slightly troubling.

As he states, the גמרא asserts that according ר' זירא the 20 אמות limit would only apply to a small Sukkah of 4x4 אמות. I know I have heard but can't remember where, that the specific issue that ר' זירא had was that the סכך needs to provide shade on the ground. If the walls of a small sukkah were 20 אמות high, there would never be shade on the ground provided by the סכך. This is certainly true. But according to the trigonometry of it, the limit should be far less than 20!

As stated by R' Ari, the highest angle the sun will reach in ארץ ישראל during this time is approximately 58°. The tan of 58° is approximately 1.6. That is to say that at a 58° angle, the opposite side (the wall) is about 1.6x the length of the adjacent (the ground.) That said, in a 4x4 sukkah, a wall of merely 6½ אמות would cause the walls to constantly cast a complete shadow on the ground and the סכך will never provide this shade. If we approach it from the other end, 20 אמות walls would prevent סכך-shade in any sukkah up to 12½ (20/1.6) אמות.

9 comments:

I think you may want to reread the post at astrotorah.weeklyshtikle.com. The issue has nothing to do with walls. The Gemara even has a case where someone builds a sukkah in a valley and states that it is fine even though the shadows are completely cast by the surrounding mountains. It does not matter if the walls are casting shadows, just like it doesn't if the mountains are. The issue is whether the schach would cast a shadow below it if there were no walls/mountains. The Ritva cited in the post over there says this directly and brings the comparison to the valley. So, no mathematical problems for Rebbi Zeirah.

As I said, I'm pretty sure I heard that understanding somewhere in ר' זירא but am not sure where. I don't think the Sukkah in a valley proves anything. That is an external factor, not where the sukkah itself is the problem. There isn't a problem, for example, if you make a sukkah where it is cloudy all the time and the sun never shines through.

My problem with not understanding ר' זירא this way is the following: This is a precise quantitative halachah but I don't see any quantitative explanation for it. All you say is that a sukkah above 20 amos does not provide ample shade. What is ample shade? What does a 19 amah sukkah provide that a 21 amah sukkah does not? And the bottom line is, such a sukkah provides absolutely no shade on the ground.

It is clear from the Ritva and seemingly from Tosefos that the focus is not the competing shadows cast by the walls, but by the schach had it been suspended somehow with no walls. The Ritva is the one to compare the walls to the mountains, as well. Essentially what the Gemara is saying, according to them, is we are looking at a 4X4 schach shadow cast on the ground. The darkness of the mild shadow cast is clearly what is being discussed and Rebbi Zeirah understands the Mishna to be saying that the level of acceptable darkness (in Erez Yisrael) is 20 amos. To paraphrase another statement from this same Rebbi Zeirah (to Rebbi Yirmiyah), do not question the exactness of the measurements given by Chazal as they are done with extreme precision. (Rosh Hashana 13a) Although the difference between 20 and 20.01 amos is extremely slight, that is where the line is drawn.

No no no. I'm not saying anything about "competing" shadows. The math is the same. Ok, assume there are no walls. My calculation shows that if the schach were a mere 7 amos above the ground of a 4x4 sukkah, there would be NO shadow cast by the schach on the ground whatsoever. The shadow cast by the schach would be cast onto the neighbouring wall. If there were no wall, it would be cast on the ground outside of the sukkah. How can we discuss the darkness of the mild shadow cast if there is no shadow to speak of? Maybe we need to discuss this in person. A simple diagram might clear this up. Ari (Brodsky), can you weigh in here (one way or the other)?

I apologize for not being clear. What I am trying to say is that the issue is not whether or not the Sukkah has enough shade, rather, whether the schach can provide the correct amount of shade EVEN THOUGH IT IS PROJECTING IT TO AN AREA THAT IS OUTSIDE THE CONFINES OF THE SUKKAH. Even though the shadow cast by the schach suspended from 6.5 amos up to 20 amos high will be projected on the ground at a distance not within the 4X4 sukkah, nevertheless, since it can produce ample shade (outside the sukkah) it is considered to be kosher schach. As such, the area directly beneath it is considered to be beneath kosher schach and can be a kosher sukkah. It does not matter that that area is not receiving shade from the schach. Of course the area to which its shade is cast is not kosher since that is not within a sukkah, just like if one were to construct a three walled sukkah and the schach-shade is cast to the direction of the open wall. Nobody would contend that that area is considered to be within the sukkah. At heights above 20 amos, with schach less than 4X4+, the shadow will be too light or too diffuse based on the square area upon which it is cast. Rebbi Zeirah is teaching what constitutes kosher schach and not what constitutes the correct shadow darkness in a sukkah.

While I do believe the above to be what the Ritva maintains, I did see that the Ran questions how shade will reach the floor from the schach with these dimesions. He concludes that Chazal must have known that somehow the schach below 20 amos affects the heat/coolness in the Sukkah while above 20 does not help enough. This would be even though no apparent shade is cast within the confines of the Sukkah.

I don't understand Ari S.'s answer to Shtikler's problem. If we are allowed to count the shade cast by the schach outside the confines of the sukka, then how does the schach of a taller sukka provide less shade than that of a shorter sukka? The surface on the ground shaded by the schach (including the ground outside the sukka, imagining the absence of walls) will always have pretty much the same area as the horizontal surface area of the schach itself.